Add the blackberries and figs to a preserving pan. Pour in the measurement water, then add the cinnamon sticks. Bring to a simmer, then simmer, uncovered, for about 10 minutes, until the fruit is just beginning to soften.
Pour the sugar into the pan and add the lemon juice. Heat gently, stirring from time to time, until the sugar has dissolved. Bring to the boil, then boil rapidly until setting point is reached (about 25 minutes). Skim with a draining spoon or stir in butter if needed.
Ladle into warm, dry jars, filling to the very top and discarding the cinnamon sticks. Cover with screw-top lids, or with waxed discs and cellophane tops secured with elastic bands. Label and leave to cool.